Subsurface geological structures can be explored with a plurality of detection methods. The detection methods can be used to infer the presence of hydrocarbons in a formation. Two examples of detection methods are seismic methods and electromagnetic methods. Seismic methods are based on the detection of elastic waves which travel through a formation. Electromagnetic methods are based on the detection of electromagnetic waves which propagate through a formation. A specific example of an electromagnetic method is the detection of marine controlled source electromagnetic (mCSEM) data. In this method, a vessel tows a dipole source through the seawater which emits a time-varying electromagnetic field into the earth. This field propagates through the formation and is detected by a plurality of sensors which are placed on the seabed or also towed behind a vessel. After the mCSEM data have been collected by the sensors, the data need to be interpreted or inverted to extract information about the formation from the data. One way of inverting the data is using a numerical model to generate a simulated data set. If the model perfectly represents the formation and the propagation of the electromagnetic field through the formation, the simulated data will be identical to the measured data. The estimation of the optimal model can be formulated as an optimisation problem in which the distance between the real data and the simulated data is minimised. However, the inversion problem is a highly non-unique problem with many possible solutions and is ill-posed in the Hadamard sense. This may lead to un-physical results of the inversion and corresponding difficulties in interpretation. In order to address this problem, a regularisation term can be included in the optimisation problem. A spatially uniform smoothing parameter may be included in the horizontal and vertical directions of the formation.
Some methods use a Tikhonov type regularization (Tikhonov, A. N. et al., 1977, Solutions of ill-posed problems; W.H. Winston and Sons.) where additional penalties are included linearly into the cost functional. The penalties are designed to enforce certain properties in the inverted resistivity profile, and narrows down the kernel of the problem. This approach was taken, for example, by Zhdanov, et al. (SEG technical Program Expanded Abstracts, 26, F65-F76). However, this term needs to be weighted properly so that the regularizing information and the data information balance each other. A different approach was taken by Abubakar et al. (Geophysics, 73, F165-F177), where the regularizing term was included multiplicatively in the cost functional, thus eliminating the estimation of the trade-off between the different terms in the cost functional. For both approaches, however, the regularization is uniform smoothing, whereby smoothing operators are applied along the horizontal and vertical directions, thus ensuring a slowly varying resistivity profile in the inverted result.